Ellipsoid shape parameters

Usage

volume(x)

# Default S3 method
volume(x)

# S3 method for class 'ellipsoid'
volume(x)

# S3 method for class 'ortensor'
volume(x)

lode(x)

# Default S3 method
lode(x)

# S3 method for class 'ellipsoid'
lode(x)

# S3 method for class 'ortensor'
lode(x)

nadai(x)

# Default S3 method
nadai(x)

# S3 method for class 'ellipsoid'
nadai(x)

# S3 method for class 'ortensor'
nadai(x)

jelinek(x)

# Default S3 method
jelinek(x)

# S3 method for class 'ellipsoid'
jelinek(x)

# S3 method for class 'ortensor'
jelinek(x)

flinn(x)

# Default S3 method
flinn(x)

# S3 method for class 'ortensor'
flinn(x)

# S3 method for class 'ellipsoid'
flinn(x)

size_invariant(x)

# Default S3 method
size_invariant(x)

# S3 method for class 'ortensor'
size_invariant(x)

# S3 method for class 'ellipsoid'
size_invariant(x)

strain_invariant(x)

# Default S3 method
strain_invariant(x)

# S3 method for class 'ortensor'
strain_invariant(x)

# S3 method for class 'ellipsoid'
strain_invariant(x)

shape_invariant(x)

# Default S3 method
shape_invariant(x)

# S3 method for class 'ortensor'
shape_invariant(x)

# S3 method for class 'ellipsoid'
shape_invariant(x)

kind(x)

# Default S3 method
kind(x)

# S3 method for class 'ortensor'
kind(x)

# S3 method for class 'ellipsoid'
kind(x)

Arguments

x: numeric. either a 3-element vector giving the ellipsoid's semi-axis lengths (in any order), an object of class "ellipsoid", or an object of class "ortensor".

Value

positive numeric

Details

The natural strain is $$\bar{\epsilon}_i = \log s_i = \log(1 + \epsilon_i)$$ with $s_1 \geq s_2 \geq s_3$ the semi-axis lengths of the ellipsoid, or $\epsilon_i$ the strains (elongation) given by $\epsilon = \frac{l-l_0}{l_0}$ (hence $s = \frac{l}{l_0}$).

Lode's parameter for strain symmetry: $$\nu = \frac{2 \bar{\epsilon}_2 - \bar{\epsilon}_1 - \bar{\epsilon}_3}{\bar{\epsilon}_1-\bar{\epsilon}_3}$$ with $\bar{\epsilon}_1 \geq \bar{\epsilon}_2 \geq \bar{\epsilon}_3$. Note that $\nu$ is undefined for spheres, but we arbitrarily declare $\nu=0$ for them (plane strain). Otherwise $-1 \geq \nu \geq 1$. $\nu=-1$ for prolate spheroids (constriction) and $\nu=1$ for oblate spheroids (flattening).

Octahedral shear strain $\bar{\epsilon}_s$ (Nádai 1963): $$\bar{\epsilon}_s = \sqrt{\frac{(\bar{\epsilon}_1 - \bar{\epsilon}_2)^2 + (\bar{\epsilon}_2 - \bar{\epsilon}_3)^2 + (\bar{\epsilon}_1 - \bar{\epsilon}_3)^2 }{3}}$$ This is the amount of strain assuming coaxial deformation (pure-shear).

Strain symmetry (Flinn 1963): $$k = \frac{s_1/s_2 - 1}{s_2/s_3 - 1}$$ The value ranges from 0 to $\infty$, and is 0 for oblate ellipsoids (flattening), 1 for plane strain and $\infty$ for prolate ellipsoids (constriction).

and strain intensity (Flinn 1963): $$d = \sqrt{(s_1/s_2 - 1)^2 + (s_2/s_3 - 1)^2}$$ This is analogous to Nadai's strain parameter.

Jelinek (1981)'s $P_j$ parameter: $$P_j = \bar{\epsilon}^{\sqrt{2 \vec{v}\cdot \vec{v}}}$$ with $\vec{v} = \bar{\epsilon}_i - \frac{\sum \bar{\epsilon}_i}{3}$

References

Flinn, Derek.(1963): "On the statistical analysis of fabric diagrams." Geological Journal 3.2: 247-253.

Lode, Walter (1926): "Versuche über den Einfluß der mittleren Hauptspannung auf das Fließen der Metalle Eisen, Kupfer und Nickel“ ("Experiments on the influence of the mean principal stress on the flow of the metals iron, copper and nickel"], Zeitschrift für Physik, vol. 36 (November), pp. 913–939, doi:10.1007/BF01400222

Nádai, A., and Hodge, P. G., Jr. (1963): "Theory of Flow and Fracture of Solids, vol. II." ASME. J. Appl. Mech. December 1963; 30(4): 640. doi:10.1115/1.3636654

Jelinek, Vit. "Characterization of the magnetic fabric of rocks." Tectonophysics 79.3-4 (1981): T63-T67.

Examples

# Generate some random orientation data
set.seed(20250411)
dat <- rvmf(100, k = 20)
s <- principal_stretch(dat)

# Volume of ellipsoid
volume(s)
#> [1] 0.171276

#  Size-related tensor invariant of ellipsoids
size_invariant(s)
#> [1] -3.196891

# Strain-related tensor invariant of ellipsoids
strain_invariant(s)
#> [1] 2.60509

# Shape-related tensor invariant of ellipsoids
shape_invariant(s)
#> [1] -0.1155451

# Lode's shape parameter for the strain symmetry ratio
lode(s)
#> [1] -0.63562

# Nadai's octahedral shear strain
nadai(s)
#> [1] 1.266186

# Jelinek Pj parameter
jelinek(s)
#> [1] 5.993392

# Flinn's intensity and symmetry parameters
flinn(s)
#> $k
#> [1] 8.243876
#> 
#> $d
#> [1] 2.975821
#> 

kind(s)
#> [1] "LLS"

# Ellipsoid data
hossack_ell <- lapply(seq.int(nrow(hossack1968)), function(i) {
  ellipsoid_from_stretch(hossack1968[i, 3], hossack1968[i, 2], hossack1968[i, 1])
})
sapply(hossack_ell, nadai)
#>  [1] 2.197985 2.462677 2.507189 1.986605 2.824592 2.421925 2.580754 2.712021
#>  [9] 2.573880 1.877145 2.556829 2.518613 2.236213 2.643440 2.652371 2.383755
#> [17] 2.361141 1.785130 2.169690 2.349450 2.362748 2.193710 2.382355 2.139940
#> [25] 1.970644 2.192330 1.945885 2.334462 1.978659 2.161851 2.792102 2.188473
#> [33] 1.895198 2.071682 2.346818 2.032335 2.653843 2.943093 2.147554 2.295867
#> [41] 2.638900 2.342928 2.626458 2.172841 2.280597 1.971667 2.457973 2.248606
#> [49] 2.032136 2.358794 2.152572 1.537893 1.142119 1.258516 1.403769 1.312832
#> [57] 1.249068 1.466010 1.599231 2.079253 1.380805 1.420466 1.690162 1.497369
#> [65] 1.443027 1.632871 1.081520 1.313677 1.499537 1.702833 1.466010 1.602093
#> [73] 1.523021 1.542330 1.814373 1.793242 1.847809 1.898982 1.969607 1.987016
#> [81] 1.680192